Whitney extension theorem

A precise statement of the theorem requires careful consideration of what it means to prescribe the derivative of a function on a closed set. One difficulty, for instance, is that closed subsets of Euclidean space in general lack a differentiable structure. The starting point, then, is an examination of the statement of Taylor's theorem.

Given a real-valued C^m function f(x) on Rⁿ, Taylor's theorem asserts that for each a, x, y ∈ Rⁿ, there is a function R_α(x,y) approaching 0 uniformly as x,y → a such that

${\displaystyle f({\mathbf {x} })=\sum _{|\alpha |\leq m}{\frac {D^{\alpha }f({\mathbf {y} })}{\alpha$ !}}\cdot ({\mathbf {x} }-{\mathbf {y} })^{\alpha }+\sum _{|\alpha |=m}R_{\alpha }({\mathbf {x} },{\mathbf {y} }){\frac {({\mathbf {x} }-{\mathbf {y} })^{\alpha }}{\alpha !}}}

(1)

where the sum is over multi-indices α.

Let f_α = D^αf for each multi-index α. Differentiating (1) with respect to x, and possibly replacing R as needed, yields

${\displaystyle f_{\alpha }({\mathbf {x} })=\sum _{|\beta |\leq m-|\alpha |}{\frac {f_{\alpha +\beta }({\mathbf {y} })}{\beta$ !}}({\mathbf {x} }-{\mathbf {y} })^{\beta }+R_{\alpha }({\mathbf {x} },{\mathbf {y} })}

(2)

where R_α is o(|x − y|^m−|α|) uniformly as x,y → a.

Note that (2) may be regarded as purely a compatibility condition between the functions f_α which must be satisfied in order for these functions to be the coefficients of the Taylor series of the function f. It is this insight which facilitates the following statement

Theorem. Suppose that f_α are a collection of functions on a closed subset A of Rⁿ for all multi-indices α with ${\displaystyle |\alpha |\leq m}$ satisfying the compatibility condition (2) at all points x, y, and a of A. Then there exists a function F(x) of class C^m such that:

1. F = f₀ on A.
2. D^αF = f_α on A.
3. F is real-analytic at every point of Rⁿ − A.

Proofs are given in the original paper of Whitney (1934), and in Malgrange (1967), Bierstone (1980) and Hörmander (1990).

Seeley (1964) proved a sharpening of the Whitney extension theorem in the special case of a half space. A smooth function on a half space R^n,+ of points where x_n ≥ 0 is a smooth function f on the interior x_n for which the derivatives ∂^α f extend to continuous functions on the half space. On the boundary x_n = 0, f restricts to smooth function. By Borel's lemma, f can be extended to a smooth function on the whole of Rⁿ. Since Borel's lemma is local in nature, the same argument shows that if Ω is a (bounded or unbounded) domain in Rⁿ with smooth boundary, then any smooth function on the closure of Ω can be extended to a smooth function on Rⁿ.

Seeley's result for a half line gives a uniform extension map

${\displaystyle \displaystyle {E:C^{\infty }(\mathbf {R} ^{+})\rightarrow C^{\infty }(\mathbf {R} ),}}$

which is linear, continuous (for the topology of uniform convergence of functions and their derivatives on compacta) and takes functions supported in [0,R] into functions supported in [−R,R]

To define E, set[1]

${\displaystyle \displaystyle {E(f)(x)=\sum _{m=1}^{\infty }a_{m}f(-b_{m}x)\varphi (-b_{m}x)\,\,\,(x<0),}}$

where φ is a smooth function of compact support on R equal to 1 near 0 and the sequences (a_m), (b_m) satisfy:

• b_m > 0 tends to ∞;
• ∑ a_m b_m^j = (−1)^j for j ≥ 0 with the sum absolutely convergent.

A solution to this system of equations can be obtained by taking b_n = 2ⁿ and seeking an entire function

${\displaystyle g(z)=\sum _{m=1}^{\infty }a_{m}z^{m}}$

such that g(2^j) = (−1)^j. That such a function can be constructed follows from the Weierstrass theorem and Mittag-Leffler theorem.[2]

It can be seen directly by setting[3]

${\displaystyle W(z)=\prod _{j\geq 1}(1-z/2^{j}),}$

an entire function with simple zeros at 2^j. The derivatives W '(2^j) are bounded above and below. Similarly the function

${\displaystyle M(z)=\sum _{j\geq 1}{(-1)^{j} \over W^{\prime }(2^{j})(z-2^{j})}}$

meromorphic with simple poles and prescribed residues at 2^j.

By construction

${\displaystyle \displaystyle {g(z)=W(z)M(z)}}$

is an entire function with the required properties.

The definition for a half space in Rⁿ by applying the operator R to the last variable x_n. Similarly, using a smooth partition of unity and a local change of variables, the result for a half space implies the existence of an analogous extending map

${\displaystyle \displaystyle {C^{\infty }({\overline {\Omega }})\rightarrow C^{\infty }(\mathbf {R} ^{n})}}$

for any domain Ω in Rⁿ with smooth boundary.

• The Kirszbraun theorem gives extensions of Lipschitz functions.

• McShane, Edward James (1934), "Extension of range of functions", Bull. Amer. Math. Soc., 40 (12): 837–842, doi:10.1090/s0002-9904-1934-05978-0, MR 1562984, Zbl 0010.34606
• Whitney, Hassler (1934), "Analytic extensions of functions defined in closed sets", Transactions of the American Mathematical Society, American Mathematical Society, 36 (1): 63–89, doi:10.2307/1989708, JSTOR 1989708
• Bierstone, Edward (1980), "Differentiable functions", Bulletin of the Brazilian Mathematical Society, 11 (2): 139–189, doi:10.1007/bf02584636
• Malgrange, Bernard (1967), Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, 3, Oxford University Press
• Seeley, R. T. (1964), "Extension of C∞ functions defined in a half space", Proc. Amer. Math. Soc., 15: 625–626, doi:10.1090/s0002-9939-1964-0165392-8
• Hörmander, Lars (1990), The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Springer-Verlag, ISBN 3-540-00662-1
• Chazarain, Jacques; Piriou, Alain (1982), Introduction to the Theory of Linear Partial Differential Equations, Studies in Mathematics and Its Applications, 14, Elsevier, ISBN 0444864520
• Ponnusamy, S.; Silverman, Herb (2006), Complex variables with applications, Birkhäuser, ISBN 0-8176-4457-1
• Fefferman, Charles (2005), "A sharp form of Whitney's extension theorem", Annals of Mathematics, 161 (1): 509–577, doi:10.4007/annals.2005.161.509, MR 2150391